The 19th century costume that Etrigan gives Batman is the same costume Batman wears in the "Gotham by Gaslight" graphic novel. This costume was designed by Mike Mignola who is also given further homage when Sherlock Holmes mentions "Mignola Chapel".
When they meet for the first time, Holmes wonders how Batman deduced his identity so quickly, to which Batman retorts "the hat." Holmes' deerstalker is never actually mentioned in any of Arthur Conan Doyle's stories, and largely became associated with Holmes because of pop-culture depictions of him; as Watson describes, Holmes was quite fashion-conscious and, given his status in London society, would not have made the faux pas of wearing such a garment.
Included as a bonus cartoon from the DC Comics Vault on the bluray release of Batman: Gotham By Gaslight (2018).